page_title: Wavelet
section_title: Comparisons with Fourier transform (continuous-time)
hierarchical_section_title: Wavelet / Comparisons with Fourier transform (continuous-time)
caption_attribution_description: English: A picture of time-frequency atoms of the short-time Fourier transform (STFT) versus the multiresolutional properties offered by the wavelet transform.
context_page_description: A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one recorded by a seismograph or heart monitor. Generally, wavelets are intentionally crafted to have specific properties that make them useful for signal processing. Using convolution, wavelets can be combined with known portions of a damaged signal to extract information from the unknown portions.
  For example, a wavelet could be created to have a frequency of Middle C and a short duration of roughly a 32nd note. If this wavelet were to be convolved with a signal created from the recording of a song, then the resulting signal would be useful for determining when the Middle C note was being played in the song. Mathematically, the wavelet will correlate with the signal if the unknown signal contains information of similar frequency. This concept of correlation is at the core of many practical applications of wavelet theory.
  As a mathematical tool, wavelets can be used to extract information from many different kinds of data, including – but not limited to – audio signals and images.
context_section_description: The wavelet transform is often compared with the Fourier transform, in which signals are represented as a sum of sinusoids. In fact, the Fourier transform can be viewed as a special case of the continuous wavelet transform with the choice of the mother wavelet
  .
  The main difference in general is that wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in frequency. The Short-time Fourier transform (STFT) is similar to the wavelet transform, in that it is also time and frequency localized, but there are issues with the frequency/time resolution trade-off.
  In particular, assuming a rectangular window region, one may think of the STFT as a transform with a slightly different kernel
  where  can often be written as , where  and u respectively denote the length and temporal offset of the windowing function. Using Parseval's theorem, one may define the wavelet's energy as
  From this, the square of the temporal support of the window offset by time u is given by
  and the square of the spectral support of the window acting on a frequency 
  Multiplication with a rectangular window in the time domain corresponds to convolution with a  function in the frequency domain, resulting in spurious ringing artifacts for short/localized temporal windows. With the continuous-time Fourier Transform,  and this convolution is with a delta function in Fourier space, resulting in the true Fourier transform of the signal . The window function may be some other apodizing filter, such as a Gaussian. The choice of windowing function will affect the approximation error relative to the true Fourier transform.
  A given resolution cell's time-bandwidth product may not be exceeded with the STFT. All STFT basis elements maintain a uniform spectral and temporal support for all temporal shifts or offsets, thereby attaining an equal resolution in time for lower and higher frequencies. The resolution is purely determined by the sampling width.
  
  In contrast, the wavelet transform's multiresolutional properties enables large temporal supports for lower frequencies while maintaining short temporal widths for higher frequencies by the scaling properties of the wavelet transform. This property extends conventional time-frequency analysis into time-scale analysis. 
  The discrete wavelet transform is less computationally complex, taking O(N) time as compared to O(N log N) for the fast Fourier transform. This computational advantage is not inherent to the transform, but reflects the choice of a logarithmic division of frequency, in contrast to the equally spaced frequency divisions of the FFT (Fast Fourier Transform) which uses the same basis functions as DFT (Discrete Fourier Transform). It is also important to note that this complexity only applies when the filter size has no relation to the signal size. A wavelet without compact support such as the Shannon wavelet would require O(N²). (For instance, a logarithmic Fourier Transform also exists with O(N) complexity, but the original signal must be sampled logarithmically in time, which is only useful for certain types of signals.)
caption: STFT time-frequency atoms (left) and DWT time-scale atoms (right). The time-frequency atoms are four different basis functions used for the STFT (i.e. four separate Fourier transforms required). The time-scale atoms of the DWT achieve small temporal widths for high frequencies and good temporal widths for low frequencies with a single transform basis set.
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